How to revive this Mogensen leather armchair?  

http://www.fredericia.com/Craft-materials/Care/Leather.aspx
On that page, it gives this caution about vegetable tanned uncoated leather:
"
Caution: Leather creams with fats should not be applied to leather that does not have surface protection to prevent the risk of blotches. Sulphated or chemical cleaning detergents must in no circumstances be used on leather furniture.
"
I think it is worth mentioned that the risk of fats is not damage to the leather. It is to avoid blotching. In other words, fats and oils will change the color of the leather, and possibly inconsistently so. I don't think you will find anyone who suggests that this damages the leather. And most people would probably say that at some number of decades fats/oils will be necessary to maintain the leather in good condition. At which point the color change is unavoidable.
The warning about detergents is of a very different category. This goes back to what I said above about leather being acidic and detergents and many chemical cleaners being basic/alkaline. Use of these chemicals will destroy the leather.

One way to tell the difference between coated leather and uncoated leather is applying water. If it soaks in then the leather is uncoated. If if beads, then it is coated.
On old leather, the coating may be worn away.
And note the leather care instructions for uncoated leather mention that water can cause blotches. Vegetable tanned leather is not very water resistant, so enough water will cause the leather to darken and harden. This is why the instructions say never to soak the surface, and to wipe down the entire surface. In other words, you are minimizing the water exposure and making sure that the color change will be minimal and uniform across the entire chair.
And I would be very careful with using soap on the leather, even if it is consistent with Fredericia's care guide. Soap is made by reacting lye with fats, and almost always there is some residual alkalinity in it from the lye. If you can find a soap that you are certain is neutral or acidic (i.e. has a PH level less than 7), then perhaps a minimal amount of it would be acceptable, but only if absolutely necessary.

Leif, thank you for this addition to your earlier comment. Yes, I saw the risk of blotches mentioned in the instructions of Fredericia for vegetable tanned leather. Indeed, applying water would be one possible way to discern the kind of leather we are dealing with. I've to think whether I can do that without causing damage to the chair.
As to the treatment of soap: I fully agree with your cautions. Much will depend on the condition of the leather. I treated the leather straps of my Runner Chair (BM 2254), which were really dry, with some blotches (dure to water, I suppose), with a little leather oil and the result was really wonderful.

There is probably a spot on the underside you could wipe with a tiny amount a water.
Even better you might start by going to a leather store and ask them for a piece of coated leather and another piece of uncoated, vegetable tanned leather (NOT the more chrome tanned leather as it responds differently to water). Then you can do some water tests on these scraps to learn what you are looking for.
After 50 years or so, with coated leather, I would be worried that the coating may have worn throughout places. If you were then to follow Fredericia's leather care guidelines, you could end up with very blotchy leather, because the applied product would enter the leather where the coating has worn off. This could require a lot of work to remedy.
(You see an analogous situation with teak, when it has been lacquered at the factory, then worn through from use/water damage, then oiled, it becomes blotchy overall as you see the the worn through areas become darker because the oil can soak into the wood).
As you noticed on the straps, oil can greatly help certain old dried leathers.
The stains probably were from water. The vegetable tanning process fills the leather with tannins, which can react, in the presence of water, with substances in the leather or in the water. Iron plus water is well known for ebonizing wood. The same reaction can happen with leather. And oxalic acid can be used to remove the stains, just like with wood.
You would do well to use distilled water on the leather, especially if you are in a place, where there are lots of dissolved minerals in the water, because lots of substances react with tannins.
I feel that Fredericia's leather care information is intended for new owners of new furniture. And recommendations for leather care might be very different after 50 years. For instance uncoated vegetable tanned leather is likely to need fats/oils after enough years of use and damage from water and the sun.
There is a lot to learn about leather. Keep reading!

Since this week, the Mogensen chair is mine. The picture I provided at the start of this post was from the former (first) owner. So I'd like to show you first the chair as I got it.

As I posted before, I've done a lot of reading about leather. The only option I could really agree about was to apply leather fat, without additives. The chair seemed to be in need of that, since especially the right sides seemed to have suffered quite a bit (due to sunlight or a heater). It felt like cardboard. So, I've been busy with feeding the leather. And the result is truly amazing in my opinion. This leather is from such an amazing quality.
It is hard to 'catch' the result in pictures, but the 'dry' look has disappeared and the wonderful 'patina' remained. I'm very happy with it. There is one point of damage, high at the back on the left side (see picture).
In short, thank you all for your kind comments and advices! And enjoy the pictures!

Looks good!